# 1.3 Order of Accuracy

## 1.3.4 Definition of Multi-Step Methods

The class of finite difference methods known as multi-step methods is one of the most widely-used approaches for solving ordinary differential equations, and forms the basis for solving partial differential equations as well.

## Definition of Multi-Step Methods

The generic form of an $$s$$-step multi-step method is,

 $v^{n+1} + \sum _{i=1}^ s \alpha _ i v^{n+1-i} = {\Delta t}\sum _{i=0}^ s \beta _ i f^{n+1-i}.$ (1.51)

A multi-step method with $$\beta _0 = 0$$ is known as an explicit method since in this case the new value $$v^{n+1}$$ can be determined as an explicit function of known values (i.e. from $$v^ i$$ and $$f_ i$$ with $$i \leq n$$). A multi-step method with $$\beta _0 \neq 0$$ is known as an implicit method since in this case the new value $$v^{n+1}$$ is an implicit function of itself through the forcing function, $$f^{n+1} = f(v^{n+1},t^{n+1})$$. We defer implicit methods to Section 1.7.1 and focus on explicit methods here.

Exercise 1 What are the non-zero $$\alpha _ i$$ and $$\beta _ i$$ for the forward Euler method?

Exercise 1

Answer:

Using the notation given above, the forward Euler method is:

 $\alpha _1 = -1 \quad \mbox{all other} \quad \alpha _ i = 0$ (1.52)
 $\beta _1 = 1 \quad \mbox{all other} \quad \beta _ i = 0$ (1.53)

Exercise 2 What are the non-zero $$\alpha _ i$$ and $$\beta _ i$$ for the Midpoint method?

Exercise 2

Answer:

Using the notation given above, the midpoint method is:

 $\alpha _2 = -1 \quad \mbox{all other} \quad \alpha _ i = 0$ (1.54)
 $\beta _1 = 2 \quad \mbox{all other} \quad \beta _ i = 0$ (1.55)